Properties of sequence of linear functionals on $BV$ with applications

TL;DR

This paper investigates the boundedness and sharpness of sequences of linear functionals on the space of functions of bounded variation ($BV$), with applications to Fourier series convergence and classical orthonormal systems.

Contribution

It introduces new conditions for boundedness of linear functionals on $BV$, enhancing understanding of their convergence properties and applications.

Findings

Proved boundedness of certain linear functional sequences under specific conditions.

Demonstrated the sharpness of these boundedness results.

Applied findings to Fourier series convergence and classical orthonormal systems.

Abstract

This paper is devoted to investigating the sequence of some linear functionals in the space $BV$ of finite variation functions. We prove that under certain conditions this sequence is bounded. We also prove that this result is sharp. In particular, the obtained results can be used to study convergence of some general Fourier series. Moreover, the obtained conditions seem to be new and useful also for classical orthonormal systems.